Abstract
Abstract This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.
Highlights
Important classes of growth models are undirected first-passage percolation (FPP) and directed last-passage percolation (LPP), where growth proceeds along optimal paths called geodesics
Our paper does not have a result for general LPP, but our proof does lay out a possible route from fairly basic properties of LPP to the non-existence of bi-infinite geodesics
The model studied is a version of nearest-neighbor directed LPP on the planar integer lattice, known as the corner growth model (CGM)
Summary
Since their inception over 50 years ago in the work of Eden [14] and Hammersley and Welsh [20], random growth models have been central drivers of the mathematical theory of spatial random processes. In the seminal paper of Johansson [23], the Tracy-Widom distribution of the limit fluctuation in LPP was proved for geometric and exponential weights This led to a large literature on exactly solvable models in the Kardar-Parisi-Zhang (KPZ) class and gave rise to a new subject, integrable probability. After the seminal paper of Cator and Groeneboom [9] on the Poissonian planar growth model, [5] derived the 1/3 shape exponent and 2/3 transversal exponent in the exponential LPP These papers point the way to a proof of the KPZ exponents under a strictly concave shape function and sufficiently mixing Busemann functions. Our paper does not have a result for general LPP, but our proof does lay out a possible route from fairly basic properties of LPP to the non-existence of bi-infinite geodesics We do this through deeper insight into the queueing picture obtained in [15]. We contrast our work with [7] in more detail
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